results, R will be arbitrarily set to a value of -3. Some results will turn out to

be independent of R. For perspective, it is a well known result of linear elasticity

theory [42] that R cannot be less than /2 for isotropic materials. Commercial

aluminum, a modestly soft and nearly isotropic metal, has an R of roughly 2

whereas GaAs, a non-isotropic semiconductor, has an R averaging close to the v2

limit.

All phonons are contemplated in the i pi -'.-...Ipi regime, by which I mean

that their wavelengths are not less than an order of magnitude smaller than the

smallest cross sectional width. I also assume that treating the material as an elastic

continuum is justified by first assuming the waveguide material has a typical inter

lattice spacing much smaller than the smallest phonon wavelength considered. As

a practical matter, this would still permit important results to apply to quantum

wires with widths on the order of a few hundred atoms. The displacements will be

assumed sufficiently small that applied elasticity theory is well within the linear

regime. For phonons that are thermally excited or scattered from interactions with

itinerate electrons, magnons, and similar particles, this is a reasonable physical

assumption consistent with remaining within the phonons' own mesoscopic regime.

Calculations and representations will be rendered within a right-handed Cartesian

2ah

--2h-

Figure 2.1: Nomenclature of Elastic Waveguide. Frequency and wave number will
be rescaled relative to h (and c8). The smaller halfwidth will be denoted hz = h.
The only geometric factor in final results will be a, the cross-section aspect ratio.

coordinate system. The long axis of the bar will be considered the x axis to

facilitate easy comparison with publications of historical significance in which this

is the more common convention. The long coordinate will be embedded along the

geometrical center of the bar. This placement of the long axis symmetrically divides

the bar. Accordingly, the bar will be transversely bounded by -hy < y < hy and

-h, < z < hz. For convenience, hz will be consistently taken as the smaller half-

width should hy / h, and h without subscript will refer to this smaller quantity. As

already indicated, a will denote the cross sectional aspect ratio so that hy = ahz.

2.2 Symbolic Consistencies and Adopted Tensor Notation

In addition to foregoing nomenclature, the base symbol for all quantities related

to elastic displacements will be the letter u. Vectors, as opposed to their compo-

nents, will be bolded. When necessary to distinguish shear versus longitudinal

displacement contributions, a parenthesized superscript will be used, as in u()

and u(s). The displacement field will be decomposed into longitudinal and shear

parts generated by a scalar potential p and a vector potential H respectively. The

Greek letter TI will be consistently chosen as a base symbol for longitudinal wave

numbers and a for shear wave numbers so that these associations can be perceived

at first glance.

Einstein notation will be used in tensor equations. Repeated indices will,

tered surfaces are often called SH waves (for "shear, with displacements horizontal

to the suin ) in contradistinction to SV waves (for -!h. ir, with dispacements

vertical to the sui .. -. ) that are coupled to longitudinal waves at surfaces.

4.

3.
44
0
.4J-
-H
2.

-H
m

0 1 2 3 4
K [hkx]

Figure 4.3: Infinite Plate SH Modes. Shear waves with displacements parallel to
the surfaces, and which vanish there, form an uncoupled propagating system in an
infinite plate. The lowest even and odd subbands are shown together.

Now, Hz, by my chosen basis representation, must have the form

Hz = B P(kx)P,(y) P (a+z)

Even & Odd SH Modes
Independent of R
. . . . . .

Thus, it is easy to see that the nontrivial solutions for this SH system follow simply

by setting the z-derivative of Pz(a+z) at z = h to zero.

P,(ah)= 0 sin(v=22 -K2)= 0 (4.25)
cos(V 2 -K2

I will make comparison to these SH solutions in the sequel. Subbands of this

solutions set are shown in Figure 4.3. They are analagous to torsional modes

of a waveguide whose dominant displacement pattern are also characterized by

displacements parallel to the surfaces.

4.5 How to Transform Superpositions

The re-derivation of the Rayleigh-Lamb equation was an exercise in organizing

the problem into an algebraic form amenable to the simple elimination of unknown

constants. The three independent boundary value equations constitute at most

three constraints. Absent superpositions, the scaling constants for the potentials

will constitute one degree of freedom for the scalar potential and as many additional

degrees of freedom as there are distinct components of the vector potential to

resolve. However, at each surface only one directional coordinate will be fixed and

so there is possibly one additional degree of freedom to be resolved with respect

to the other. The Rayleigh-Lamb scenario reduces enough degrees of freedom

to balance the constraints. Specifically, restricting the scenario to plane waves

eliminates the directional degree of freedom at each surface and reduces the number

of constants from a maximum of four to a manageable two. With the nontrivial

constraints reduced to the same number, a solution follows.

In the rectangular waveguide case, the directional degree of freedom at each sur-

face persists by the fact that plane waves no longer suffice. Barring some fortuity,

the constraint equations will go to six-three for each surface-with no eliminations.

A question arises: how many independent vector potential components must there

be and thus how many degrees of freedom due to them? Two directional degrees

of freedom plus the scalar plus at least one vector potential component makes

the minimum degrees of freedom to be four. If I couple the .,.1] i:ent sides, the

independent constraints may be reduced and if I include additional vector potential

components I can increase the degrees of freedom. But as I have demonstrated

Preservation of the ability to transform into delta functions, despite divergence

of the Fourier transform and other issues, can be secured by a simple expedient.

Instead of the general Fourier transform, I use a -IiIl!!., '1" transform which is

valid only on exponentials (though without regard to whether the wave numbers are

real or imaginary) and which avoids the divergence problem by virtue of the details

of its defined domain. Moreover, since the domain of functions to be transformed

is explicitly constrained by the physical boundaries of the sample, there is nothing

illogical or restrictive in defining the applicable domain for the transform to include

only exponentials defined on the coordinate intervals that measure the sample and

thus there need not be any concern over periodic extension. The transform to be

used then has a simple definition determined by the element mapping

aeV 2a6(7 r) (4.26)

-hq < q < hq (q = x, y, z) 7,i TE R U 9 a E C

The factor of 2 is a convenience to dispose of the factor I in the exponential

representation of the sine and cosine.

It is almost self-evident that the transform is 1-1 between the set of exponentials

and set of delta functions so restricted. Since a can be zero, an additive identity

exists on each side and we have an isomorphism between two groups. Because the

range of r] and 7 is defined to encompass both real and pure imaginary values, the

transform operates without difficulty on any combination of exponentials with real

or imaginary wave numbers.

It is troublesome to write down an integral form of this transform which

smoothly adapts to whether the argument is real versus imaginary and which

limits itself to the coordinate boundaries. Of course, the underlying mechanism is

a trivial Fourier transform. Fortunately, since the domain of functions is strictly

limited and the element mapping from that domain to the transform domain is

clear and unambiguous, the transform can be performed without difficulty. It is

inil i. -I i1- therefore, that the inverse transform can be trivially written down in

an integral form that is not troublesome and which applies adaptively to transforms

involving imaginary delta arguments as well as real ones. One example would be

1 rX=+o \ =+ioo
f (y) {2A f (A) e CAYdA + e AdA
2 VX-oo J-ioo

Of course, this is only valid for f(A) that are produced by the -:iiIlIII. ,I" trans-

form in the first place.

The left or right hand side of any boundary value constraint will involve one or

more sums in the following general form (which omits common factors of P (k x)

which are also subject to differentiation):

Saj f (p, pj) P, (pj qi)P2 (p q2) (4.27)
3

Here, I have generalized the various cases:

ql, q2 stand for distinct coordinates y or z;

P1, P2 are distinct function variables, or derivatives of them: Py or P,;

p will be an r or a for representations of the scalar or component of the vector

potential respectively;

pt is an abbreviation for the conjugate wave number based upon the applicable
velocity: p* = /z2/c-k2-p2 p+ = /2/c2-k2-p2;

f (pj, pt) will be a prefactor resulting from one or more derivatives taken. These

will ahlv--, be a single product or sums of products of pj and/or p>. To

characterize the effect of transforms on the boundary expressions, it will be

sufficient to contemplate f(pj, p1) as being a single such product since a sum

of such terms can be distributed to produce sums of summations.

By summarizing below how the ilphi.d" transform affects boundary value

terms generically defined by equation (4.27), it will be possible to immediately

write down the transforms of the actual boundary conditions without elaboration.

First, with y -- A chosen to make the example concrete, consider the general effect

of the transform on a function variable.

1 1
y{P( y)} 6(A ) + 6(A + ) (4.28)
-i\ -1

I can now write down the transform of equation (4.27) with respect to ql 7.

Note that 7 stands for whichever transform dimension variable is matched to ql.

My convention henceforth will be that y -- A and z -- In the rendering of

transforms, ql and q2 could be either y or z, though alv--iv- distinct in a given case.

While the reader is presumed capable of writing down Fourier transforms of
sines and cosines on his/her own, the usual results are somewhat simplified and
adjusted in this case by virtue of the stipulation that superpositions will alv--
be chosen to involve only positive (albeit possibly imaginary) values of whatever
wave number variable p designates. The result, therefore, is that only one of the
two delta terms survive in each case and the specific form of the result depends
upon the sign of the transform variable in a way that can be neatly summarized.
So, for convenience I list the results in detail:

u-fl { Ej aj f (pj, pj) K(pji qi)Pqp (p q2)}

a f (, 7t) 1 ( ) P (7t q2) for 7 > 0 (4.29)
Pq1

aj f(O, Ot) 1 2 6p) (t q2) for 0 (4.30)
-i p 0

ajf( 7) 6 (171- ) (7q2) for/71<10 (4.31 )
-i -1

Similarly, I can now write down the transform of equation (4.27) with respect to
q2. There are again the same three cases depending upon the sign or 7.

q2-7 { Ej a f (pj, pj) Pq (pj qi)lPq(p q2)}

Sa f (Qt) 1 ( -P ) q 1(7t ) for7>0 (4.32)
{f2

a f(0t,0) 1 2 6(0o-p) P,(0tqi) for7 0 (4.33)
I P2 IP2

Zaj f(t7 7{) } 1 } P p) (7t qi) for 7 < 0 (4.34)
S-i -1 2

Having been meticulous in motivating, then ju -I iii:- and now demonstrating

the effects of this -iiiiphjii' i transform. It can be drastically simplified in practice

with the following observations:

The prior stipulation that expansions will only be over positive, though possibly

including imaginary, wave numbers has allowed the results of transforms in the

context of the problem to be more easily summarized in terms of single delta

functions. One additional fact can now literally trivialize the use of this transform

in practice. Namely, the fact that any given derivation takes place in the context of

a specific parity pattern guarantees that within any derivation the parity pattern

on each side of a boundary equation will be identical. This parity agreement

guarantees that when either coordinate is transformed, the P function transformed

on each side will be the same. That being the case, all of the prefactors which

depend on the parity of P will cancel between the sides in all cases where that P

function is common to all additive terms. In general, this condition nearly alv-

fulfilled. Moreover, with those distinctions gone, an examination of equations

(4.29-4.34) will reveal that replacing 7 with |7| throughout is sufficient to cover all

cases. Therefore, as a practical matter, the only rule that will be needed is just

Replace every Pq(pq q) to be transformed with 6(171 pq).

It will not matter whether pq is a wave number or its conjugate.

It will not matter whether pq is real or imaginary (it is guaranteed to

be positive).

The triviality of this rule is a direct consequence of having imposed a meticulous

series of specific choices. It is neither fortuitous nor could it have been readily

anticipated that it would reduce to this.

69

Once a boundary equation that has had a superposition substituted into it

becomes transformed in this way, it becomes an equality in terms of the transform

variable. Given the behavior of delta functions, each value of the transform variable

will, on each side of the summation, select out either a specific term of the sum,

or be identically zero. This collapses the equality of functions of sums into a

constraint between components of sums from each side. Correlating transforms

over y with those over z is a remaining issue, but how this must be done will be

developed in the derivations which follow.

CHAPTER 5
DERIVING NORMAL MODES OF PROPAGATION

5.1 General Considerations

The full boundary conditions revealed generically by equations (2.22) infer that

all three components of the shear vector potential are mixed together in satisfying

the stress-free surface boundary constraint. If, however, I restrict attention to just

those modes which propagate, it is not immediately clear whether all three vector

components are needed. Some inspiration can be drawn from the Rayleigh-Lamb

derivation of section 4.4. That derivation involves an infinite plate bounded in

the z directions and only requires the Hy component of the vector potential. The

intimation is that bounding also in the y directions might simply invoke the need

for the H, component, but there is no a priori reason to expect to need an H,

component as well.

The foregoing motivates an attempt to find the essential relationship between

H, and the other vector components which participate in satisfying the boundary

conditions along the surface of a waveguide in which normal modes propagate in

the x direction. Unless some constraint can be found that eliminates components or

establishes some dependency among them, there will be more degrees of freedom in

the propagating problem then constraints available to resolve them. I thus proceed

to investigate this relationship among components in a way that is independent of

the boundary conditions per se so as to confidently narrow the approaches used in

solving the boundary problems.

5.2 Acoustic Poynting Vector of a Normal Mode

Propagating modes carry energy. In analogy with electrodynamics, there will

be a vector that indicates both the direction and magnitude of the energy flux. This

Finally, the last two terms on the right of equation (5.11) contribute the

following terms to the x component of the Poynting vector:

-4 [H ],,i + (H[x,,]p,xy H, .,xz)]

-2 [p,xp,,x + (pP,yP,xy + P,zPz)] (5.18)

It is easily checked that the desired antisymmetry is preserved and that, again, the

vanishing either of Hx alone or Hy and Hz together does not change this result.

The conclusion is that, when assembling a propagating normal mode, the shear

contribution must be made out of components for which all the Hx's are zero, or

for which all the Hy and Hz parts are zero. In considering how to represent the

shear superpositions of a propagating normal mode, there is no case in which a

superposition for H, will be mixed with ones for Hy and H,. This removes at least

one degree of freedom from the problem.

5.3 Propagating Modes Involving Hy, H, Shear

5.3.1 Deriving the Frequency Equations

The ease with which the Rayleigh-Lamb solution is derived inspires a deriva-

tion that follows the same pattern. Armed with the conclusion that H, cannot

even be accommodated in a normal mode solution that also includes Hy and H,

components, I proceed to derive the spectrum of propagating modes with H, = 0.

Accordingly, from equations (2.22), the boundary conditions at z = h, with

S- z, p -- x, p -+ y, and Hx = 0 become

3 k2j p + 2,p,,

I,zy

- [(Hy,zz Hy,zz) + Hz,zy]

+ (Hz,zz Hyy)

(5.19)

As a reminder, under the basis rules devised for this problem (see section 4.2), the

representations of potentials will have the following forms:

S= Px(k x) di Py(li y) P,(T* z) with T1*
i

0

H = P,(k x) E, aP,( y-)P(a z) with

Ej bjP,(y7 y)P(aj+ z)

2/ _2 ck2 i 2

o( = w2/c2 2 _- 2
a+ S

2Hy,x

Substituting into the first boundary condition of equations (5.19) I obtain

di (Ok} + 2(l)2)P (riy) P,(Tl h,)

2k {+1 P,(Gj y) Pz (j h,) (5.20)
-1 3 -1

The -iip!.-" transform devised in section 4.5 is then applied so that Py(rTiy)

6(IA\ ,y) and Py(7j y) -) 6(IA oj). By choosing a value Ao of the transform

variable A such that Ao E {(r} i {nj}, the sums on both sides collapse leaving the
following equality:

do (Ok2 + 2(A )2) P(A h ) 2k { o oA Pz(A, h,) (5.21)
-1 I-1

This provides one constraint on possible combinations of (w, Ao) at a given value
of k. Here and in subsequent steps, Ao =- Ao .

Repeating this process for the second boundary condition in equations (5.19),
the transformed version of the second constraint becomes

do k A: Pz(A h,)
P P

-P,(Af h) -ao(k2 (A 2) bo Ao(A) (5.22)
-1 1
Py P_

Substituting into the third boundary condition in equations (5.19), the trans-
formed result is

do -AoA 1 P}(A h)

-k } bo A -ao Ao P (A}+ h,) (5.23)
2 -1 1 1
P. Pz Py

Equations (5.22) and (5.23) can be reconciled into one constraint by finding a
relationship between ao and bo that renders them equivalent. Dividing the two
equations will eliminate the transcendental terms and some common factors, leav-
ing

ao ({ k2 (+A)2) + bo A, A+

-k--2 P (5.24)

bo A+ ao Ao

Solving this for the required relationship between coefficients yields

bo -o A[2 (5.25)

Substitution of equation (5.25) into either equation (5.22) or (5.23) to eliminate bo
will produce the following result:

The transformed first boundary condition from equations (5.29), before appli-

cation of the delta functions, will be

ZdQ 2k + 2T/)P(/i ah,) 6( ) =

2k E-j Py (j ah,) (af+ -/) (5.30)
1 3 -1

To link the .,11i i:ent surface conditions, I rely upon the fact that each root of

equation (5.27) necessarily corresponds to the existence of specific elements of the

support sets {T1} and {oaj}. In fact, if Ao is, with some value of w, a root of equation

(5.27), it is solely because 3Iro E {ql} and 3o0 E {aj} such that Ao = ro = ao.

Now, the left sum in equation (5.30) a will necessarily encounter Ao = To.

Suppose, then, that I contemplate the value of p corresponding to To = Ao for

which the delta function on the left of equation (5.30) is nonzero. Obviously, it will

be A*. If, for that value of p, the right hand side of equations (5.30) is not trivially

zero for the same value of p, there must exist some ao such that a = = A .

In the alternative, I could first contemplate a value of p for which the argument

of the delta function on the right of equation (5.30) is zero. Obviously, it will be

A+. If, for that value of p, the left hand side of equations (5.30) is not trivially

zero for the same value of p, there must exist some fll such that Tl = p = A+.

The upshot of this reasoning is that I can connect the transforms of the two

sets of boundary conditions by contemplating simultaneous (w, Ao) roots of both

of them. To write the transformed boundary conditions for the .,1i ,i:ent side in

terms of Ao, I require either that

p --> A

so that, by operation the delta function, the wave number variables take on values:

rli Tlo Ao

di do which will be eliminated

aoj o- i.e. presumed to exist

ai al unknown, but to be eliminated

J+ A

S- (A)+ (5.31)

or I require that

P--> A+

so that, by operation the delta function, the wave number variables alternatively

take on values:

Jj g o = Ao

bj bo which will be eliminated

+
cry -- Ao

Tli ll i.e. presumed to exist

ai al unknown, but to be eliminated

i (A)* (5.32)

I will name the first of these alternatives (i.e., equations (5.31 )) "L-Conjugation"

since it is premised on equating the longitudinal conjugation of the z-surface

solution with the y-surface solution. The second alternative (i.e., equations (5.32))

"S-Conjugation" since it is premised on equating the shear conjugation of the

z-surface solution with the y-surface solution.

Because I have stipulated that all members of the support sets are positive

and because only positive square roots are used, all of the preceding mappings are

guaranteed to be unambiguous. The reader is invited to review the definitions of

conjugation denoted with and + superscripts which were introduced in connec-

tion with equations (4.8) and (4.10). From those definitions, it can be noted that

these conjugations have the property

(A*)- A (A+)+ A

A detailed expansion of the final relation in equations (5.31) is

(A)+ (A*)2 \A+2 A2 ()2 (5.33)

and of the final relation in equations (5.32) is

(A) -) -2 (A+)2 )2 + ( 2 (5.34)

These expansions illustrate the general rule that L-Conjugation and S-Conju-
gation solutions are related by straightforward substitutions of variables. It will
thus be sufficient to complete details of the ongoing derivation for the L-Conjugate
case and then state the analogous results for the S-Conjugate case. Applying
L-Conjugation to equation (5.30), and after collapsing the sums, the result is

-do (pk + 2A ) P,(Aoah,) = 2k b1 (A*)+ P1[(A)+ ahz] (5.35)

Similarly, the second and third boundary conditions (5.29) under L-Conjugation
can eventually be put into the following forms:

1 -1
01k \ \\ }Py(Aoah )

iP,[(A/)+ ah] -bi[k2 ((A )+)2]+ a1 A (A)+ (5.36)
1 -1
2 ~L\/o \'\

do0AAo{ }{ Py(Aoahz)
1t 1
[y bl P

Sk at (A *) b1 A:* P[(A*)+ ahz] (5.37)
-1 1 1

Dividing equations (5.36) and (5.37) to put one al in terms of b1, I obtain

-1 -1 t (A)+Ao*
a = bi )(5.38)
k2 + 2
ht t 0iM^

It may be noted that equation (5.38) is not identical to (5.25). This highlights the
fact that al and bl are expected to be distinct from ao and bo. If equation (5.38)
is substituted into either equation (5.36) or equation (5.36) to eliminate al, the
result is identical, to wit:

Had anyone been insightful enough to anticipate that coupled modes of a

rectangular waveguide could be characterized by a coincidence of Rayleigh-Lamb

solutions, proving that it was so would have remained as elusive as history shows

the main problem to have been. Moreover, there are aspects of the result which,

had it been somehow forseen as possible, would have argued against believing it.

The main impediment would have been that there is an intrinsic interference built

into the result which precludes boundary satisfaction at both surfaces without

the rest of the superposition. Although the elegant-looking result involves a

coincidence with the full superposition-it is still not the full superposition. What

is intriguing is that I do not need to have a full description of the superposition in

order to find the eigenspectrum.

The derivation tells us that any superposition of shear and longitudinal compo-

nents that satisfy boundary conditions at .,.i i,:ent sides must include a particular

combination of components that, in a partial sense, mimic a Rayleigh-Lamb wave

system. The L-Conjugation and S-Conjugation cases are merely two different v-i-,

of realizing this. Figures 5.1 and 5.2 illustrate the essence of these alternatives.

They imply two recipes for building the superpositions.

88

z=h

r7 =r 17

long shear

..+
II
r +
C b
II
S on,= (7*)+

shear L-Conj

II

Figure 5.1: Illustrating the L-Conjugation Case for Modal Solutions. Shown are the
relationships among longitudinal and shear wave vector components of the defining
physical waves which must be among those making up the total superposition
needed for a solution. The common x directional component (k) is normal to the
page.

The recipe implied by L-Conjugation begins with a longitudinal wave at a

desired k value. Conceptually, one could imagine starting out with some r] and

some u close to a modal solution. Add a shear wave with the same fixed k.

The polarization of this initial shear contribution is such that shear displacement

is not parallel to the sides. Now adjust u until the Rayleigh-Lamb equation is

satisfied with respect to the z = h, sides. There will be a range of o's for which

Rayleigh-Lamb can be satisfied at these parallel sides. For each possible U the bulk

dispersion relations will fix if* and a+ wave vector components pointing along the

z directions. Now add a second ("conj- i, i. ) shear wave at the same k and close

89

z= h

cr I rj = cr

long
shear *
b
II

+
b

II

7conj= ("+)* S-
S-Conj

long

b
II
11F
0

Figure 5.2: Illustrating the S-Conjugation Case for Modal Solutions. Shown are the
relationships among longitudinal and shear wave vector components of the defining
physical waves which must be among those making up the total superposition
needed for a solution. The common x directional component (k) is normal to the
page.

to the w implied by the process so far. The polarization of this shear wave should

also result in a displacement not parallel to the sides. The initial a+cj is set equal

to *. Just as the first shear wave had a wave vector component common with the

foundation longitudinal wave along the y direction, this conjugate shear wave has

a wave vector component common with the foundation longitudinal wave along the

z direction. Bulk dispersion will fix the value of ocomj. Now adjust w over the range

of values that continue to satisfy Rayleigh-Lamb at the z = h, surfaces until one

is found for which Rayleigh-Lamb is also satisfied for the foundation longitudinal

wave and the just-added conjugate shear wave at the y = hy surfaces. When

such an u is found, it defines an eigenfrequency of a propagating system at the set

value of k. There will be as irn Ii such o/s as there are subbands.

The recipe implied by S-Conjugation is procedurally the same as the recipe

for L-Conjugation except that the foundation wave is a shear wave instead of a

longitudinal one and two longitudinal ones are added instead of two shear waves.

The recipes each focus on combining waves in total disregard of their reflections

at .Ii] ,' ent surfaces. Satisfaction of Rayleigh-Lamb incorporates reflections only at